I resist as incorrect every time I have a chance. It only became the British short form fairly recently. And, to anyone who uses the singular short form "math", it makes a suggestion that mathematical conclusions are open to being overturned the way scientific conclusions are. Since mathematical conclusions are final, there is only one math.

It's probably a hypercorrection because the original term is mathematics, and therefore its short form should be plural. No, if I want to make it short - I'll leave off as many letters as I want to. From this point on, a capital M refers to the field of mathematics.

Infinity is not a number. It only exists as a formal notion to indicate lack of bound. One can talk about a "point at infinity", but to do so, one must adjoin to the set of numbers an object that is not numeric, but which only exists as a formal notion. Therefore, any set which contains a point at infinity is not a set of numbers. Yes, this is a "no true Scotsman" argument, but it has to be so because it is an argument about definitions.

makes no sense from a mathematical perspective. Does an infinite number of photons exist? Probably not (I am not a physicist). But the question is not that. The question is whether an infinite number of photons can fit into a space occupied by a laser. If that number has no bound, then infinitely many photons can fit there.

Ok, maybe I should put it this way. You may be able to draw a formal distinction between "countably many" and "any number of", but you cannot draw that distinction logically; certainly not in the context of counting.

Eventually the laser energy will create a black hole, provided some other exotic effect doesn't occur first. Realisitcally though it's not possible to attain those kinds of photon densities (nothing can reflect anywhere close to well enough for starters).

They did. Photons will gain an increasing energy density, and a corresponding increasing pressure, which both have an impact on the gravitational field and, if high enough in a concentrated area, could indeed cause the appearance of a black hole. That does not mean they have "mass". The photon energy is basically E=hbar*omega or E=pc, depending how you want to write it. Here omega is the angular frequency, p the momentum, while hbar and c are Planck's constant and the speed of light respectively.

It does not mean they have "rest mass", but they do actually have "mass", m=E/c2, that should create gravity around the photon ( if gravity is such as thing that gets created and is present around an object, and it's not just a straight "instantaneous action at a distance" between two objects. now my head is starting to hurt)

Eventually the laser energy will create a black hole, provided some other exotic effect doesn't occur first.

That's a limit on energy density, not total energy in the laser. In principle you could use a very WIDE laser opterating below the black-hole thrshold and focus the beam externally (which, if it's powerful enough, it might do eventually, by self-gravitation, after leaving the cavity, even if the cavity geometry made it emit a colimated, rather than a converging, beam.) Thus, making a kugelblitz with a (very wide) laser might be theoretically possible (if "some other exotic effect" didn't make the required laser cavity to wide to be physically realizable).

I'd imagine "Some other exotic effects" might include the electric field component of the coherent light becoming strong enough to polarize the vacuum and create particle-antiparticle pairs from multiple photons, dissipating their energy, somewhere WAY below the threshold of gravitic-collapse effects. So you'd need a REALLY WIDE laser and REALLY GOOD optics to make your external-to-the-laser black hole.

Of course the question, being phrased in terms of Bose-Einstein vs. Fermi-Dirac statistics and "infinite" energy was really about energy density in the cavity - just poorly phrased. So you answered the question that was REALLY being asked.

E = mc^2 specifically applies only to objects that have nonzero mass and are at rest with respect to the observer. Photons are massless and move at the speed of light.

The general equation is E = sqrt((mc^2)^2 + (pc)^2) for rest mass m and momentum p. If a particle has mass and is at rest, then p=0 so E=mc^2. If a particle is massless, then m=0 so E=pc.

(The "m" here refers to rest mass m0, not the "relativistic mass" m* which is defined as m* = m0 / sqrt(1-(vc)^2)). Relativistic mass is best thought of as a fake concept to hide the ugly sqrt denominator. People can imagine things getting heavier when they're moving, and can keep saying "Einstein discovered E=mc^2". But it still has division-by-zero problems with massless particles, and things don't really "get heavier" when they move, so if you try to avoid thinking in terms of m* you won't get as confused. Neither m nor m* makes E=mc^2 work with photons.

Imagine if a bundle of photons could gather and form a "black hole". The hole and its event horizon would be constrained to move at the speed of light, which you can't, since you have mass. so you might easily escape its event horizon- you wouldn't have time to fall in before the thing was gone. Real black holes have mass and don't move at the speed of light relative to anybody.

... nothing that we have technological access to at the moment. You might manage to reflect light with a high-enough magnetic field, but getting it flat enough to form a lasing cavity isn't going to be easy.

But long before that happens the question is if the laser can remain a laser.

A laser needs some kind of nonlinearity in the medium. Any nonlinearity introduces a scale. So the real question is: At which power does of-resonant driving cause transitions (e.g. Landau-Zener) or of-resonant shifts (Stark shift) and can you actually theoretically contruct a medium which fulfills the criteria to serve as a lasing medium for an arbitrary large scale of power?

As a starting point for an examination of such questions i recomment the Quantum Optics Toolbox for Matlab by Sze Tan.

"Update: After a conversation with Chad Orzel, it looks like although there's no limit to the photon energy you can produce, you will at some point--above about 1 MeV in photon energy--start spontaneously producing matter-antimatter pairs of particles whenever your photon interacts with a reflective surface. So at extremely high photon energies, your laser light begins to resemble a matter-antimatter thermal bath rather than merely coherent light."

So it wouldn't be possible to construct a reflective surface that's not solid in the traditional sense but a reactive field of energy, which would guide the process at the point of interaction - to bypass the limit?

I just searched for an answer to this question. Seems that pair generation by irradiation of matter (e.g. a mirror) is shown experimentally and can reach quite high intensities:http://journals.aps.org/prl/ab... [aps.org]
Generation in vacuum though seems to be shown only in models until now:http://iopscience.iop.org/0295... [iop.org] http://journals.aps.org/pra/ab... [aps.org]
Seems that the reaction rate is much lower, so maybe this is not a limiting factor for building a laser.
Normally high intensities are achieved by building a pulsed laser. This produces a beam of laser pulses, which is then focussed into a tiny spot. Intensities in this spot can be alot higher than inside the laser cavity. You could achieve higher laser intensities just by building a larger laser (like http://en.wikipedia.org/wiki/N... [wikipedia.org]
).
Inside the laser cavity intensities are normally limited by the effects of nonlinear optics (
http://en.wikipedia.org/wiki/N... [wikipedia.org]
), which occur in all kinds of matter.

This effect will not only kill at high energies but at high intensity too. With a high enough intensity you can have multi-photon interactions to achieve the same total energy. However pair production is not the only process you have to worry about compton scattering [wikipedia.org] will occur as well. This will impose an intensity limit well below pair production energies.

Essentially reflected photons will have less energy than incident photons and as the energy increases so too does this energy difference. It is cause

For normal matter — things like protons, neutrons and electrons — there's a fundamental limit to the number of particles you can fit into a given region of space thanks to the Pauli exclusion principle.

Wrong, unless you assume space is discretized, which might happen around Planck's length, but has never been proven theoretically nor experimentally.

The power generation isn't one billionth as hard as - how the hell do you get that energy (presumably electrical) to the device in the first place in a usable format? You can alway just build a nuclear fusion plant, then another, then another, then another, then another, in close proximity.

But somewhere, somehow, you have to transport or convert that amount of energy in a non-light way, which is going to involve some humungously gigantic amount of heat on a physical component, or some monstrously huge devi

But somewhere, somehow, you have to transport or convert that amount of energy in a non-light way, which is going to involve some humungously gigantic amount of heat on a physical component, or some monstrously huge device to attempt to dissipate the heat.

Why? If raw power is all you're concerned with, just use gamma rays and/or neutrons from an exploding nuclear bomb to pump your lasing media. Sure, you'll only get a single pulse, but that's all you'll need.

The problems of generation are solveable - we just need a way to harness something like a Sun (e.g. Dyson spheres). The problem you really have is how do you concentrate that energy onto a point such that it generates a laser?

Well, if you have a Dyson sphere you can shoot a laser from every point on the surface facing the target, they won't be perfectly aligned but they will pass through the same volume of space, like a magnifying glass effect with lasers. That should make a rather nice bug zapper.

Billions and billions of years ago, even before lord Xenu, there was a scientist who pulled this off.

Blext Telfrawd, an A type Hixoid, did get an infinite number of protons into a finite space. Then the containment field faltered, obliterating the iteration of his universe..

Most historians agree this was tragic for it ended his universe, and created one with Justin Bieber. Sentients who were able to achieve trans-dimensional universital access, send a message to you from the past: It's just too risky to repeat the so called "Bieber Event",

The main problem with testing this is "how does one generate infinite or near-infinite energy" to power something like this?

Of course, if we've answered that, we're ALREADY in a place where we've either wiped ourselves out (accidentally or otherwise), or we've basically solved the greatest real-world problem in the history of humanity.

The energy of a photon is characterized by its wavelength. In a laser, the wavelength is constant. You have a large amount of photons which are coherent but at an almost single wavelength. When the article is talking about 1 MeV, it falsely interprets this as if the laser is emitting a single photon at 1 MeV. That is not what happen. It emits many photons in coherence which the sum of energy of all the individual photons will reach 1 MeV or more. Each photon cannot create an electron-positron pair and all photons collectively cannot create an electron-positron pair.

A 1 MeV photon would be a gamma ray photon and it is not true at all, your laser doesn't change its wavelenght as more more "energy" is emitted. In fact, we should instead talk about the power of the laser rather than its energy. The power being the amount of energy emitted by unit of time.

There is no apparent upper limit to the energy of a photon. The galaxy Markarian 501 emits photons in the teraelectronvolt (TeV) range.

The question here is about intensity. The relativistic energy-moment dispersion, E^2=(mc^2)^2+(pc)^2, which applies to all on-shell particles, has a gap when m>0. This gap, which is about 1 MeV for electrons and positrons, can be overcome when the electric field (generated by a sufficient number of photons, irrespective of their energy) approaches the Schwinger limit of about 1.3 x 10^18 V/m. At this point, virtual electron-positron pairs can be created in abundance because the mass gap has been overcome, and electromagnetism then becomes non-linear. Pumping in more photons after this simply creates more virtual e-p pairs.

When the article is talking about 1 MeV, it falsely interprets this as if the laser is emitting a single photon at 1 MeV. That is not what happen

He is indeed talking about 1 MeV per photon. He's discussing the theoretical limits of photon power density in a hypothetical gamma-ray laser with an adjustable wavelength. An ordinary laser pointer stores more than 1 MeV of energy in its lasing cavity, although a physicist would not typically use eV to describe the combined energy of a light beam.

He starts with the idea of an ordinary laser. Those are not even in the X-ray range, nevermind the MeV gamma-ray range. Then he wants to 'compress' the lasing cavity to *ahem* reach black-hole level of energy densities. While you can transfer energy to the radiation field (thus shifting up photon energies from the visible/UV range) you'll need a HECK of a fast compression to reach the electron-positron generation threshold. So that's nonsense.

Then he wants to 'compress' the lasing cavity to *ahem* reach black-hole level of energy densities.

It seems pretty clear to me—I took that same first course—that a neutrino is just a white hole (moving at the speed of light) made up of photons which such a strong self-interaction they can't escape from themselves and thus refuse to interact with much of anything else.

This all seemed to fit with the gravitational contribution of the EM Stress Energy Tensor until I saw a post from Lubos on Stackex

If you can channel the energy of a fusion explosion into many lasing-while-ionizing rods (think "Real Genius"s death ray laser, but MUCH larger) you could pack so many X-Ray photons into a burst that the impact (momentum transfer) alone destroys the target's armor, at least according to David Weber.

There was an article from 2010 that talked about the theoretical limit to laser beam energy. From the article:

"At high laser intensities interaction of the created electron and positron with the laser field can lead to production of multiple new particles and thus to formation of an avalanche-like electromagnetic cascade"

As a certain energy density, the radiation pressure from the photons will be stronger than the tensile strength of the optical cavity, and the laser will blow apart. In astronomy, a similar limit is called the Eddington limit, so this is really the Eddington limit for a laser.

The radiation pressure is (ignoring all factors of 2 or cos(incidence)) E / c. A tensile limit, T, of 500 mega pascals (reasonable for steel) thus would imply an energy intensity of c T, or 1.5 x 10^17 Watts/m^2. If the total cavity had an area of 1 m^2, then that's ~ 10^17 Watts.

Note that it is common in pulsed lasers to have a lot of energy in a very short pulse (so the actual power during the pulse is very high). If your pulses were a microsecond in length, then the Eddington limit per pulse would be about 10^11 Joules, equivalent to 24 tons of TNT.

The first part is irrelevant, as he asked the limit for "a cavity" (i.e., inside a specific sized object), and not on the target, but the second is not. You can, in fact, make your laser from a bomb and evade such limits. This was basically Edward Tellers idea for gamma ray lasers for anti-missile defense.

At high enough energies particles are spontaneously created. They in turn will obey Pauli Exclusion (at least if they have spin I think). So enough photons and you make matter that will prevent you from making more particles ie pumping more energy into the space.

Not sure. There is weirdness in particle physics where you are able to break the rules as long as the duration is short enough that you never could measure it. Sometimes those "broken rules" states are simplify calculations. If you pumped in enough energy I'd guess you could get a big bang. Another option would be that the expansion of the edge of the high energy region combined with the reduction of the speed of light in this new material would be such that you stop increasing the energy density of the cen

"Physics" is not just one thing anymore. The guy writing TFA, Ethan Siegel, is a bonified professional physicist. Reading the comments, you can see he just didn't know this one thing as well as he thought. How does that happen?

I don't know that there's any physicist going through training today or in the last 20 years who really understands "all" of physics.

Physics PhDs learn most of physics up to about 1910 (even that is a stretch, but at least the complete fields up to that point are introduced and sketched out), and the next 100 years are based on your specialty. The limits of energy density for photons are usually in this realm of "introduced only if directly important to your specialty."

It's up to the individual to fill in the gaps after formal classes, and it can be very hard to figure out what you don't know. It's particularly hard because of the oversimplified way physics is generally taught in undergrad, even to physics majors. Your old reference books may not actually be correct. I'm sure I've got a physics textbook around which claims almost exactly what Ethan said in his blog; the "why" of pair generation is just too distracting.

Bose-Einstein Condensate!
In more detail, fermions cannot be crammed together but in certain conditions, Bosons can. Photons are a type of Boson but not the only one. The Pauli exclusion principle does not apply to Bosons! Looks like a non-specialist needs to read some books on this concept. I won't even go into deeper details without this point being crystal clear!